Angles in a Triangle Worksheets - Math Monks - Free Printable

Let's solve each problem step by step using the properties of interior and exterior angles of a triangle:

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🔷 Key Concepts:

1. Sum of interior angles in a triangle = 180°
2. Exterior angle = sum of two opposite interior angles
3. An exterior angle is formed when one side of a triangle is extended.

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Now, let's go through each question:

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1. ∠ECD = ?

Given:
- ∠D = 21°
- ∠E = 31°

We need to find the exterior angle at C, ∠ECD.

Using the exterior angle theorem:
> Exterior angle = sum of two non-adjacent interior angles

So,
$$
∠ECD = ∠D + ∠E = 21° + 31° = \boxed{52^\circ}
$$

✔ Answer: 52°

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2. ∠TQR = ?

Given:
- ∠T = 55°
- ∠R = 39°

We are asked for ∠TQR — this is the interior angle at Q.

Use triangle angle sum:
$$
∠Q = 180° - ∠T - ∠R = 180° - 55° - 39° = \boxed{86^\circ}
$$

✔ Answer: 86°

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3. ∠MNP = ?

Given:
- ∠O = 77°
- ∠P = 77°

We are looking for ∠MNP — that’s the angle at N.

But wait: ∠MNP is the exterior angle at N, since point M is outside.

Wait — look at the diagram: Triangle O-N-P, with angle at N labeled as ∠MNP (the exterior angle).

So we need to find the exterior angle at N.

First, find the interior angle at N:
$$
∠N = 180° - 77° - 77° = 26°
$$

Then, the exterior angle at N is supplementary to the interior angle:
$$
∠MNP = 180° - 26° = \boxed{154^\circ}
$$

✔ Answer: 154°

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4. ∠MOQ = ?

Triangle P-Q-M, with:
- ∠P = 87°
- ∠Q = 63°

We want ∠MOQ — this is the exterior angle at Q, because side QO is extended.

So use exterior angle theorem:
$$
∠MOQ = ∠P + ∠M
$$

But we don’t know ∠M yet.

Find ∠M first:
$$
∠M = 180° - 87° - 63° = 30°
$$

Then:
$$
∠MOQ = ∠P + ∠M = 87° + 30° = \boxed{117^\circ}
$$

✔ Answer: 117°

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5. ∠NPM = ?

Triangle M-N-P:
- ∠M = 60°
- ∠N = 60°

We are to find ∠NPM — that’s the angle at P.

So:
$$
∠P = 180° - 60° - 60° = \boxed{60^\circ}
$$

This is an equilateral triangle!

✔ Answer: 60°

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6. ∠HFX = ?

Triangle H-W-F, with:
- ∠H = 39°
- ∠W = 26°

We want ∠HFX — this is the exterior angle at F, formed by extending side HF to X.

So use exterior angle theorem:
$$
∠HFX = ∠H + ∠W = 39° + 26° = \boxed{65^\circ}
$$

✔ Answer: 65°

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7. ∠RPQ = ?

Triangle P-Q-R:
- ∠Q = 51°
- ∠R = 46°

We want ∠RPQ — that’s the angle at P.

So:
$$
∠P = 180° - 51° - 46° = \boxed{83^\circ}
$$

✔ Answer: 83°

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8. XAC = ?

Triangle ABC:
- ∠B = 47°
- ∠C = 60°

We are to find ∠XAC — this is the exterior angle at A, formed by extending side BA to X.

So, use exterior angle theorem:
$$
∠XAC = ∠B + ∠C = 47° + 60° = \boxed{107^\circ}
$$

✔ Answer: 107°

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✔ Final Answers:

| Problem | Answer |
|--------|--------|
| 1 | ∠ECD = 52° |
| 2 | ∠TQR = 86° |
| 3 | ∠MNP = 154° |
| 4 | ∠MOQ = 117° |
| 5 | ∠NPM = 60° |
| 6 | ∠HFX = 65° |
| 7 | ∠RPQ = 83° |
| 8 | ∠XAC = 107° |

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Let me know if you'd like this formatted into a printable answer sheet!